Goncharov's direct-sum conjecture for multiple zeta values

For each natural number $w$, let $Z_w$ be the $\mathbb Q$-vector space generated by multiple zeta values of weight $w$, let $Z_0=\mathbb Q$, and let $Z_\bullet=\bigoplus_{w\geqslant0}Z_w$ be the formal graded MZV algebra. Let $d:Z_\bullet\to\mathbb R$ be the natural ring homomorphism that is the identity on every $Z_w$. Goncharov's direct-sum conjecture. The homomorphism $d$ is injective; equivalently, there are no non-trivial $\mathbb Q$-linear relations among multiple zeta values of different weights. The conjecture would imply transcendence of all multiple zeta values, while the source records only partial results and says that proving the direct-sum assertion appears difficult.